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Cusps of Hyperbolic Knots

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Cusps of Hyperbolic Knots Preliminaries Main Theorems Constructions Hyperbolic geometry Cusps of Hyperbolic Knots Shruthi Sridhar Cornell University [email protected] Collaborators: Michael Moore, Rose Kaplan-Kelly, Brandon Shapiro, Joshua Wakefield Research from SMALL 2016 Preliminaries Main Theorems Constructions Hyperbolic geometry 1 Preliminaries 2 Main Theorems 3 Constructions 4 Hyperbolic geometry Preliminaries Main Theorems Constructions Hyperbolic geometry Hyperbolic Knots Definition A hyperbolic knot or link is a knot or link whose complement in S3 is a 3-manifold that admits a complete hyperbolic structure. This gives us a very useful invariant for hyperbolic knots: Volume (V) of the hyperbolic knot complement. Figure 8 Knot 5 Chain Volume=2.0298... Volume=10.149..... Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Definition A Cusp of a knot K in S3 is defined as an open neighborhood of the knot intersected with the knot complement: S3 \ K Thus, a cusp is a subset of the knot complement. Definition The Cusp Volume of a hyperbolic knot is the volume of the maximal cusp in the hyperbolic knot complement. Since the cusp is a subset of the knot complement, the cusp volume of a knot is always less than or equal the volume of the knot complement. Fact √ The cusp volume of the figure-8 knot is 3 Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Volume The maximal cusp is the cusp of a knot expanded as much as possible until the cusp intersects itself. Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Volume The maximal cusp is the cusp of a knot expanded as much as possible until the cusp intersects itself. Definition The Cusp Volume of a hyperbolic knot is the volume of the maximal cusp in the hyperbolic knot complement. Since the cusp is a subset of the knot complement, the cusp volume of a knot is always less than or equal the volume of the knot complement. Fact √ The cusp volume of the figure-8 knot is 3 Fact The optimal cusp volume of links will be achieved by first, maximizing a particular cusp till it touches itself, then a particular second cusp till it touches itself or the first cusp and so on... Fact The optimal cusp volume√ of the minimally twisted 5-chain is 5 3√where√ the√ individual√ √ 3 3 3 3 cusps have volumes: 4 3, 4 , 4 , 4 , 4 respectively. Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Volume of Links How should we expand cusps in a link to achieve maximal volume? Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Volume of Links How should we expand cusps in a link to achieve maximal volume? Fact The optimal cusp volume of links will be achieved by first, maximizing a particular cusp till it touches itself, then a particular second cusp till it touches itself or the first cusp and so on... Fact The optimal cusp volume√ of the minimally twisted 5-chain is 5 3√where√ the√ individual√ √ 3 3 3 3 cusps have volumes: 4 3, 4 , 4 , 4 , 4 respectively. Preliminaries Main Theorems Constructions Hyperbolic geometry Cusp Density Definition CV Cusp Density (Dc ) of a knot or link is the ratio: V where CV is the total cusp volume and V is the hyperbolic volume of the complement. Volume=2.0298...√ Volume=10.149...√ Cusp Volume= 3 Cusp Volume = 5 3 Cusp Density=0.853 Cusp Density=0.853 Theorem (SMALL 2016) For any x ∈ [0, 0.853...], there exist hyperbolic link complements with cusp density arbitrarily close to x. Theorem (SMALL 2016) For any x ∈ [0, 0.6826...], there exist hyperbolic knot complements with cusp density arbitrarily close to x. Preliminaries Main Theorems Constructions Hyperbolic geometry Main Theorems Fact (B¨or¨oczky 1978) The highest cusp density a hyperbolic manifold can have is 0.853..., the cusp density of the figure-8 knot and the minimally twisted 5-chain. Preliminaries Main Theorems Constructions Hyperbolic geometry Main Theorems Fact (B¨or¨oczky 1978) The highest cusp density a hyperbolic manifold can have is 0.853..., the cusp density of the figure-8 knot and the minimally twisted 5-chain. Theorem (SMALL 2016) For any x ∈ [0, 0.853...], there exist hyperbolic link complements with cusp density arbitrarily close to x. Theorem (SMALL 2016) For any x ∈ [0, 0.6826...], there exist hyperbolic knot complements with cusp density arbitrarily close to x. Preliminaries Main Theorems Constructions Hyperbolic geometry Constructions Three constructions provide essential elements to the proofs: Dehn Filling Cyclic covers Belted sums Cq C Fact As q → ∞, the Volume and Cusp T T volume approaches that of the original link before Dehn filling Vq V Preliminaries Main Theorems Constructions Hyperbolic geometry Dehn Filling Definition (1, q) Dehn filling on an unknotted component of a link complement gives a link complement of the original link, without the trivial component, and the strands through it twisted q times. Cq C T T Vq V Preliminaries Main Theorems Constructions Hyperbolic geometry Dehn Filling Definition (1, q) Dehn filling on an unknotted component of a link complement gives a link complement of the original link, without the trivial component, and the strands through it twisted q times. Fact As q → ∞, the Volume and Cusp volume approaches that of the original link before Dehn filling We know that volume of an n−fold cover Ln is n times the volume of the original link L. V (Ln) = nV (L) In fact, the same holds for cusp volumes and we can see this as a multiplication of the fundamental domain. CV (Ln) = nCV (L) Preliminaries Main Theorems Constructions Hyperbolic geometry n-fold Covers The following construction is taking ”n-fold cyclic covers” of a tangle across a twice punctured disc. Twisted Borromean Rings 3-fold cover Preliminaries Main Theorems Constructions Hyperbolic geometry n-fold Covers The following construction is taking ”n-fold cyclic covers” of a tangle across a twice punctured disc. Twisted Borromean Rings 3-fold cover We know that volume of an n−fold cover Ln is n times the volume of the original link L. V (Ln) = nV (L) In fact, the same holds for cusp volumes and we can see this as a multiplication of the fundamental domain. CV (Ln) = nCV (L) V (L1) + V (L2) = V (L1+2) We know that volume of a belted sum is the sum of the volumes of the original 2 links. What happens to cusp volumes? Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums The following construction is taking a belted sum of links cross a twice punctured disc T2 T1 T2 T1 L1 L2 Belted Sum: L1+2 Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums The following construction is taking a belted sum of links cross a twice punctured disc T2 T1 T2 T1 L1 L2 Belted Sum: L1+2 V (L1) + V (L2) = V (L1+2) We know that volume of a belted sum is the sum of the volumes of the original 2 links. What happens to cusp volumes? The boundary of a cusp is a torus. The meridian is marked in yellow and the longitude in red. Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Cusp volume doesn’t add so nicely. L1 L2 L1+2 ∗ ∗Under theT absence2 of poking Theorem (SMALLT1 16) T2 2 T1 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2 m1 and m2 are meridian lengths of the maximal cusps of L1 and L2 Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Cusp volume doesn’t add so nicely. L1 L2 L1+2 ∗ ∗Under theT absence2 of poking Theorem (SMALLT1 16) T2 2 T1 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2 m1 and m2 are meridian lengths of the maximal cusps of L1 and L2 The boundary of a cusp is a torus. The meridian is marked in yellow and the longitude in red. Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Heuristic Proof: Preliminaries Main Theorems Constructions Hyperbolic geometry Belted Sums Heuristic Proof: Volume of Cusp ∝ meridian2 2 new m1 RCV (L2 ) = RCV (L2) m2 2 m1 RCV (L1+2) = RCV (L1) + RCV (L2) m2 2 m1 RCV = aRCV1 + b RCV2 and V = aV1 + bV2 m2 Finally, perform high Dehn filling to the unknotted component to 2 a m1 ( )RCV1+ RCV2 b m2 get a knot with cusp density close to a . ( b )V1+V2 Choosing the right links gives us cusp density dense in [0, 0.6826..] Preliminaries Main Theorems Constructions Hyperbolic geometry Finishing the proof We find two tangles augmented by a link with volumes V1 and V2, restricted cusp volume RCV1 and RCV2 and meridians m1, m2. We take 0a0 fold covers of one and belt sum with a 0b0 fold cover of the other to get a link. Preliminaries Main Theorems Constructions Hyperbolic geometry Finishing the proof We find two tangles augmented by a link with volumes V1 and V2, restricted cusp volume RCV1 and RCV2 and meridians m1, m2. We take 0a0 fold covers of one and belt sum with a 0b0 fold cover of the other to get a link. 2 m1 RCV = aRCV1 + b RCV2 and V = aV1 + bV2 m2 Finally, perform high Dehn filling to the unknotted component to 2 a m1 ( )RCV1+ RCV2 b m2 get a knot with cusp density close to a . ( b )V1+V2 Choosing the right links gives us cusp density dense in [0, 0.6826..] Fact Cusps of hyperbolic knots lift to disjoint unions of horoballs in the upper half model of hyperbolic space Preliminaries Main Theorems Constructions Hyperbolic geometry Covering space H3 Fact Complements of hyperbolic knots lift to several copies of 3 polyhedra in H Preliminaries Main Theorems Constructions Hyperbolic geometry Covering space H3 Fact Complements of hyperbolic knots lift to several copies of 3 polyhedra in H Fact Cusps of hyperbolic knots lift to disjoint unions of horoballs in the upper half model of hyperbolic space We first look at the boundary of the cusp.
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